mutigrid methods for solving differential equations ferien akademie 05 – veselin dikov

Mutigrid Methods for Solving Differential Equations

Ferien Akademie ’05 – Veselin Dikov

Multigrid Methods

Ferien Akademie ’05 Veselin Dikov

Agenda

● Model problem

● Relaxation. Smoothing property

● Elements of Multigrid

● Multigrid schemes

Multigrid Methods Model Problem


0)1()0(

0,10),()()("

uu

xxfxuxu

● Discretization in n points, step h = 1/n

● 1D boundary problem of steady state temperature a long a uniform rod

0

,11,2

0

2

11

n

jjjjj

vv

njfvh

vvv

Multigrid Methods Model Problem


● Stencil notation

● Av = f, where

2

2

2

2

21

1..

...

121

12

1

h

h

h

hA

)121(1 2

2 h

hA

Tnvvv 11 ,..., Tnfff 11 ,..., and

● A is Symmetric positive definite

Multigrid Methods


Agenda

● Model problem




Multigrid Methods Iterative Methods


● Jacobi and Gauss-Seidel methods

● Iterative vs Direct methods

● Smoothing property

More about iterative methods

Multigrid Methods Smoothing Property


● Error along the domain

After 35 sweeps with weighted Jacobi

Error was smoothed



● Smoothing property explained in four steps

1. Fourier modes

11,0,sin,

nknjn

jkv jk

● k – wave number




1. Fourier modes

k = 1 k = 2

k = 7 k = 12




1. Fourier modes

● smooth modes - 2

1n

k

● oscillatory modes - 12

nkn




1. Fourier modes

● f = 0, σ = 0 Au = 0

2. Modified model problem

● exact solution: u = 0

● error: e = u – v = -v

we can trace the error!




1. Fourier modes


● wJacobi step

3. Weighted Jacobi relaxation

cvRv kk )()1(

● error )0()( eRe mm

21

1..

...

121

12

22

IAIR




1. Fourier modes


● we relax with wJacobi with ω = 2/3 on initial guesses respectively:


6)0(

3)0(

1)0( , vvandvvvv

4. Three experiments

e

# iterations




1. Fourier modes


● repeat the experiment with:

ω = 2/3 and initial guess


631)0(

2

1

2

1vvvv

4. Three experiments

e

# iterations




1. Fourier modes



4. Three experiments● Explanation

● Rω has the same eigenvectors as A and they are the same as the wave vectors

● Recall that for the error e(m) = Rme(0)

● Eigenvalues of Rω ?




1. Fourier modes



4. Three experiments● Explanation

wavenumber k

Eig

enva

lue




1. Fourier modes



4. Three experiments● Explanation● Smoothing property

● Fast damping of oscillatory error modes

● Common for all iterative methods

● How to overcome the bad performance effect over smooth error modes?

Multigrid Methods


Agenda

● Model problem




Multigrid Methods Elements of Multigrid


● Element I: A smooth wave looks more oscillatory on a coarser grid

● Aliasing: k looks like (n-k)



● Element II: Nested Iterations

coarsest grid

finest grid

Relax

Relax

Relax

transfer the coarse grid result to the finer grid for the initial guess

● Problems?



● Element III: Correction scheme

● Residual equation: Ae = r

● The scheme:

» Relax on Au = f on to obtain an approximation .» Compute .» Relax on Ae = r on to obtain an approximation to the error, .» Correct the approximation .

hhv

hAvfr h2

he2

hhh evv 2



● Element IV: Interpolation and restriction

● Interpolation :

● Restriction :

● Variational property:

hhI 2

.12

0,2

1

,

21

212

22

n

jvvv

vv

hj

hj

hj

hj

hj

hhI2

.22 h

jhj vv Injection:

.12

1,24

112212

2 n

jvvvv hj

hj

hj

hjFull weighting:

RcIcITh

hhh ,2

2

Multigrid Methods


Agenda

● Model problem




Multigrid Methods Two-Grid


● Two-Grid = Corr.Scheme+Interpolation+Restriction

» Relax times on on with initial guess

» Compute and restrict .

» Solve on .

» Interpolate and correct .

» Relax times on on with initial guess

hhh fvMGv ,

1 hhh fuA hhv

hhhh vAfr hhh

h rIr 22 hhh reA 222 h2

hhh

h eIe 22 hhh evv

2 hhh fuA hhv

Multigrid Methods Two-Grid -> V-Cycle


● V-Cycle = Recursive Two-Grid Scheme

● Two-Grid Scheme

V-Cycle W-Cycle

Multigrid Methods Full Multigrid(FMG)


● FMG = V-Cycle + nested iterations

FMG

Multigrid Methods Costs


● V-Cycle costs

d

d

Lddd nn

21

2

2

1...

2

1

2

112

2

WUdLdd

21

2

2

1...

2

1

2

112

2Computational cost

Storage

● FMG computational costs

WUdLddd 2221

2

2

1...

2

1

2

11

21

2

● Speedup because working on smaller domains

mutigrid methods for solving differential equations ferien akademie 05 – veselin dikov

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